Hyperbolic Functions for GCSE CCEA Mathematics: Key Points | GCSE CCEA 数学：双曲函数考点精讲

📚 Hyperbolic Functions for GCSE CCEA Mathematics: Key Points | GCSE CCEA 数学：双曲函数考点精讲

Hyperbolic functions appear in the CCEA GCSE Further Mathematics specification as an extension of the exponential function. They model many real-world phenomena, from hanging cables to special relativity, and provide a powerful set of tools for calculus. Mastering their definitions, graphs, identities, differentiation and integration is essential for achieving top marks.

双曲函数作为指数函数的延伸，出现在 CCEA GCSE 进阶数学大纲中。它们可以模拟许多现实世界现象，从悬垂的电缆到狭义相对论，并为微积分提供了一套强大的工具。掌握它们的定义、图形、恒等式、求导和积分是取得高分的关键。

1. Definition of Hyperbolic Functions | 双曲函数的定义

The two fundamental hyperbolic functions are defined in terms of the exponential function eˣ. The hyperbolic sine, sinh x, and hyperbolic cosine, cosh x, are given by

两个基本的双曲函数用指数函数 eˣ 定义。双曲正弦 sinh x 和双曲余弦 cosh x 由下式给出

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

From these, the hyperbolic tangent, tanh x, is obtained as the ratio of sinh x to cosh x.

由此，双曲正切 tanh x 被定义为 sinh x 与 cosh x 的比值。

tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)

These definitions mirror the circular functions but lack the alternating signs, leading to fundamentally different properties when squared.

这些定义类似于圆函数，但没有交替的正负号，导致在平方时性质完全不同。

2. Graph of y = sinh x | y = sinh x 的图形

The graph of y = sinh x is an odd function that passes through the origin and increases without bound in both directions. It resembles a skewed cubic curve but actually grows exponentially for large |x|.

y = sinh x 的图形是一个奇函数，经过原点，并在两个方向上都无限增加。它类似于扭曲的三次曲线，但在 |x| 很大时实际呈指数增长。

Key features: it is symmetric about the origin, sinh(0) = 0, and its gradient at the origin is 1 because the derivative cosh x equals 1 when x = 0. As x → ∞, sinh x → ½ eˣ, and as x → −∞, sinh x → −½ e⁻ˣ, so the graph approaches the exponential curves but is not asymptotic to any straight line.

关键特征：它关于原点对称，sinh(0) = 0，在原点处的梯度为 1，因为导数 cosh x 在 x = 0 时等于 1。当 x → ∞ 时，sinh x → ½ eˣ，当 x → −∞ 时，sinh x → −½ e⁻ˣ，因此图形趋近于指数曲线，但没有任何直线渐近线。

3. Graph of y = cosh x | y = cosh x 的图形

The curve y = cosh x is an even function with a minimum point at (0, 1). It is shaped like a catenary – the curve formed by a hanging chain. Unlike sinh x, cosh x is always positive and is never less than 1.

曲线 y = cosh x 是一个偶函数，最低点为 (0, 1)。它的形状像一条悬链线——悬挂的链条形成的曲线。与 sinh x 不同，cosh x 始终为正且从不小于 1。

Because cosh x = ½(eˣ + e⁻ˣ), it grows exponentially as x → ±∞. The graph is symmetric about the y‑axis, and for large |x| the term e⁻|ˣ| becomes negligible, so cosh x ≈ ½ e|ˣ|. The gradient on the left is negative, zero at x = 0, and positive on the right, reflecting the derivative sinh x.

因为 cosh x = ½(eˣ + e⁻ˣ)，当 x → ±∞ 时呈指数增长。图形关于 y 轴对称，当 |x| 很大时，项 e⁻|ˣ| 变得可忽略不计，所以 cosh x ≈ ½ e|ˣ|。左侧梯度为负，x = 0 时为零，右侧为正，这正反映了它的导数 sinh x。

4. Graph of y = tanh x | y = tanh x 的图形

The hyperbolic tangent is an odd function that tends to horizontal asymptotes y = 1 as x → ∞ and y = −1 as x → −∞. It passes through the origin and has gradient 1 there.

双曲正切是一个奇函数，当 x → ∞ 时趋向水平渐近线 y = 1，当 x → −∞ 时趋向 y = −1。它经过原点，且此处梯度为 1。

Since tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ), dividing numerator and denominator by eˣ gives tanh x = (1 − e⁻²ˣ) / (1 + e⁻²ˣ), which clearly shows the limits ±1. The graph is steepest at the origin and flattens out towards the asymptotes, never exceeding 1 in absolute value.

由于 tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)，分子分母同除以 eˣ 得到 tanh x = (1 − e⁻²ˣ) / (1 + e⁻²ˣ)，这清晰地表明极限为 ±1。图形在原点处最陡峭，并向渐近线趋于平坦，绝对值永远不会超过 1。

5. The Fundamental Identity cosh²x − sinh²x = 1 | 基本恒等式 cosh²x − sinh²x = 1

The most important hyperbolic identity is cosh²x − sinh²x = 1, which is analogous to cos²θ + sin²θ = 1 but has a minus sign. It is proved directly from the exponential definitions.

最重要的双曲恒等式是 cosh²x − sinh²x = 1，它类似于 cos²θ + sin²θ = 1，但符号为减号。该式可直接从指数定义证明。

Calculate cosh²x − sinh²x = [(eˣ + e⁻ˣ)/2]² − [(eˣ − e⁻ˣ)/2]² = ¼[(e²ˣ + 2 + e⁻²ˣ) − (e²ˣ − 2 + e⁻²ˣ)] = ¼ × 4 = 1. This identity underpins many calculations, such as solving equations and simplifying expressions.

计算 cosh²x − sinh²x = [(eˣ + e⁻ˣ)/2]² − [(eˣ − e⁻ˣ)/2]² = ¼[(e²ˣ + 2 + e⁻²ˣ) − (e²ˣ − 2 + e⁻²ˣ)] = ¼ × 4 = 1。这个恒等式是许多计算的基础，例如解方程和化简表达式。

6. Other Useful Hyperbolic Identities | 其他有用的双曲恒等式

Just as trigonometric functions have double‑angle formulas, hyperbolic functions have similar identities, often with sign changes. Two key ones are the double‑argument hyperbolic identities:

正如三角函数有倍角公式，双曲函数也有类似的恒等式，但通常伴随符号变化。两个关键的倍角双曲恒等式是：

sinh(2x) = 2 sinh x cosh x

cosh(2x) = cosh²x + sinh²x = 2 cosh²x − 1 = 1 + 2 sinh²x

These can be derived by writing sinh(2x) and cosh(2x) in terms of e²ˣ and e⁻²ˣ, or by applying the addition formulas. You may also need to relate tanh x to sech x, where sech x = 1 / cosh x, giving the identity 1 − tanh²x = sech²x.

这些可以通过将 sinh(2x) 和 cosh(2x) 用 e²ˣ 和 e⁻²ˣ 表示，或者应用加法公式来推导。有时还需要将 tanh x 与 sech x 联系起来，其中 sech x = 1 / cosh x，从而得到恒等式 1 − tanh²x = sech²x。

7. Solving Equations with Hyperbolic Functions | 解含有双曲函数的方程

Equations involving hyperbolic functions often require rewriting them in terms of eˣ or using identities. For example, to solve sinh x = 3 cosh x, divide by cosh x to obtain tanh x = 3, then express tanh x in exponential form.

含有双曲函数的方程通常需要将其重新写成 eˣ 的形式或使用恒等式。例如，解 sinh x = 3 cosh x，两边除以 cosh x 可得 tanh x = 3，然后将 tanh x 写为指数形式。

Thus (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ) = 3 ⇒ eˣ − e⁻ˣ = 3eˣ + 3e⁻ˣ ⇒ −2eˣ = 4e⁻ˣ ⇒ e²ˣ = −2, which has no real solution. Another type uses cosh²x − sinh²x = 1: given sinh x = 2, find cosh x. Since cosh²x = 1 + sinh²x = 5, cosh x = √5 (positive because cosh x ≥ 1).

于是 (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ) = 3 ⇒ eˣ − e⁻ˣ = 3eˣ + 3e⁻ˣ ⇒ −2eˣ = 4e⁻ˣ ⇒ e²ˣ = −2，无实数解。另一种类型利用 cosh²x − sinh²x = 1：例如已知 sinh x = 2，求 cosh x。由于 cosh²x = 1 + sinh²x = 5，cosh x = √5（取正值，因为 cosh x ≥ 1）。

8. Differentiation of Hyperbolic Functions | 双曲函数的求导

The derivatives of hyperbolic functions are pleasingly cyclic and mirror the trigonometric derivatives without the negative signs.

双曲函数的导数具有优美的循环性，并且与三角函数的导数相似，但没有负号。

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech²x

These can be proven directly from the exponential definitions. For instance, d/dx [½(eˣ − e⁻ˣ)] = ½(eˣ + e⁻ˣ) = cosh x. When differentiating composite functions, the chain rule applies: d/dx [sinh(ax+b)] = a cosh(ax+b). Being able to differentiate confidently is crucial for tangent equations and related rates problems.

这些可直接从指数定义证明。例如，d/dx [½(eˣ − e⁻ˣ)] = ½(eˣ + e⁻ˣ) = cosh x。在对复合函数求导时，需应用链式法则：d/dx [sinh(ax+b)] = a cosh(ax+b)。能够熟练求导对于求切线方程和相关变化率问题至关重要。

9. Integration of Hyperbolic Functions | 双曲函数的积分

Integrating hyperbolic functions is equally straightforward. Since differentiation and integration are inverse processes, the integrals follow immediately from the derivatives.

双曲函数的积分同样简单。由于求导和积分互为逆运算，积分公式可直接从导数得出。

∫ sinh x dx = cosh x + C

∫ cosh x dx = sinh x + C

∫ sech²x dx = tanh x + C

For tanh x, rewriting it as sinh x / cosh x leads to the logarithmic integral: ∫ tanh x dx = ln|cosh x| + C. When the integrand involves a linear function of x, remember to divide by the coefficient: ∫ sinh(ax+b) dx = (1/a) cosh(ax+b) + C. Definite integrals can then be computed to find areas under curves or between curves.

对于 tanh x，将其写成 sinh x / cosh x 可得到对数积分：∫ tanh x dx = ln|cosh x| + C。当被积函数含有 x 的线性函数时，注意除以该系数：∫ sinh(ax+b) dx = (1/a) cosh(ax+b) + C。此后可计算定积分，以求出曲线下方或曲线之间的面积。

10. Worked Example and Exam Tips | 典型例题与考试技巧

Let’s work through a typical GCSE Further Maths problem: Find the equation of the tangent to the curve y = 2 cosh x at the point where x = ln 2. First, find the y‑coordinate: cosh(ln 2) = ½(eˡⁿ² + e⁻ˡⁿ²) = ½(2 + ½) = 5/4, so y = 5/2. The derivative dy/dx = 2 sinh x, and at x = ln 2, sinh(ln 2) = ½(2 − ½) = 3/4, giving gradient 2 × 3/4 = 3/2. The tangent equation is y − 5/2 = (3/2)(x − ln 2).

让我们看一个典型的 GCSE 进阶数学问题：求曲线 y = 2 cosh x 在 x = ln 2 处的切线方程。首先，求 y 坐标：cosh(ln 2) = ½(eˡⁿ² + e⁻ˡⁿ²) = ½(2 + ½) = 5/4，因此 y = 5/2。导数为 dy/dx = 2 sinh x，在 x = ln 2 时，sinh(ln 2) = ½(2 − ½) = 3/4，梯度为 2 × 3/4 = 3/2。切线方程为 y − 5/2 = (3/2)(x − ln 2)。

When revising, practice converting between exponential and hyperbolic forms, sketch graphs clearly showing asymptotes and intercepts, and always double‑check the sign when using identities. In integration, watch for the need to use the identity 1 − tanh²x = sech²x to recognise standard forms. A solid command of hyperbolic functions will give you a real advantage in the Further Maths exam.

复习时，要练习在指数形式和双曲形式之间转换，清晰地画出具有渐近线和截距的图形，并在使用恒等式时反复检查符号。在积分中，注意可能需要使用恒等式 1 − tanh²x = sech²x 来识别标准形式。扎实掌握双曲函数将使你在进阶数学考试中真正占据优势。

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Hyperbolic Functions for GCSE CCEA Mathematics: Key Points | GCSE CCEA 数学：双曲函数 考点精讲